Algebra Examples

$4 x - \frac{1}{3} \leq 1 + \frac{x}{6}$

Step 1

Move all terms containing $x$ to the left side of the inequality.

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Step 1.1

Subtract $\frac{x}{6}$ from both sides of the inequality.

$4 x - \frac{1}{3} - \frac{x}{6} \leq 1$

Step 1.2

To write $4 x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$4 x \cdot \frac{6}{6} - \frac{x}{6} - \frac{1}{3} \leq 1$

Step 1.3

Combine $4 x$ and $\frac{6}{6}$ .

$\frac{4 x \cdot 6}{6} - \frac{x}{6} - \frac{1}{3} \leq 1$

Step 1.4

Combine the numerators over the common denominator.

$\frac{4 x \cdot 6 - x}{6} - \frac{1}{3} \leq 1$

Step 1.5

Simplify each term.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Factor $x$ out of $4 x \cdot 6 - x$ .

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Step 1.5.1.1.1

Factor $x$ out of $4 x \cdot 6$ .

$\frac{x (4 \cdot 6) - x}{6} - \frac{1}{3} \leq 1$

Step 1.5.1.1.2

Factor $x$ out of $- x$ .

$\frac{x (4 \cdot 6) + x \cdot - 1}{6} - \frac{1}{3} \leq 1$

Step 1.5.1.1.3

Factor $x$ out of $x (4 \cdot 6) + x \cdot - 1$ .

$\frac{x (4 \cdot 6 - 1)}{6} - \frac{1}{3} \leq 1$

Step 1.5.1.2

Multiply $4$ by $6$ .

$\frac{x (24 - 1)}{6} - \frac{1}{3} \leq 1$

Step 1.5.1.3

Subtract $1$ from $24$ .

$\frac{x \cdot 23}{6} - \frac{1}{3} \leq 1$

Step 1.5.2

Move $23$ to the left of $x$ .

$\frac{23 x}{6} - \frac{1}{3} \leq 1$

Step 2

Move all terms not containing $x$ to the right side of the inequality.

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Step 2.1

Add $\frac{1}{3}$ to both sides of the inequality.

$\frac{23 x}{6} \leq 1 + \frac{1}{3}$

Step 2.2

Write $1$ as a fraction with a common denominator.

$\frac{23 x}{6} \leq \frac{3}{3} + \frac{1}{3}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{23 x}{6} \leq \frac{3 + 1}{3}$

Step 2.4

Add $3$ and $1$ .

$\frac{23 x}{6} \leq \frac{4}{3}$

Step 3

Multiply both sides by $6$ .

$\frac{23 x}{6} \cdot 6 \leq \frac{4}{3} \cdot 6$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $6$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{23 x}{6} \cdot 6 \leq \frac{4}{3} \cdot 6$

Step 4.1.1.2

Rewrite the expression.

$23 x \leq \frac{4}{3} \cdot 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{4}{3} \cdot 6$ .

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Factor $3$ out of $6$ .

$23 x \leq \frac{4}{3} \cdot (3 (2))$

Step 4.2.1.1.2

Cancel the common factor.

$23 x \leq \frac{4}{3} \cdot (3 \cdot 2)$

Step 4.2.1.1.3

Rewrite the expression.

$23 x \leq 4 \cdot 2$

Step 4.2.1.2

Multiply $4$ by $2$ .

$23 x \leq 8$

Step 5

Divide each term in $23 x \leq 8$ by $23$ and simplify.

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Step 5.1

Divide each term in $23 x \leq 8$ by $23$ .

$\frac{23 x}{23} \leq \frac{8}{23}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $23$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{23 x}{23} \leq \frac{8}{23}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{8}{23}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{8}{23}$

Interval Notation:

$(- \infty, \frac{8}{23}]$